Theorem nrexdv 2563

Description: Deduction adding restricted existential quantifier to negated wff. (Contributed by NM, 16-Oct-2003.)

Hypothesis

Ref	Expression
nrexdv.1	|- ( ( ph /\ x e. A ) -> -. ps )

Assertion

Ref	Expression
nrexdv	|- ( ph -> -. E. x e. A ps )

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   A( x)

Proof of Theorem nrexdv

Step	Hyp	Ref	Expression
1		nrexdv.1	. . 3 |- ( ( ph /\ x e. A ) -> -. ps )
2	1	ralrimiva 2543	. 2 |- ( ph -> A. x e. A -. ps )
3		ralnex 2458	. 2 |- ( A. x e. A -. ps <-> -. E. x e. A ps )
4	2, 3	sylib 121	1 |- ( ph -> -. E. x e. A ps )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   E.wrex 2449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-4 1503  ax-17 1519

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-ral 2453  df-rex 2454

This theorem is referenced by:  ltpopr  7557  cauappcvgprlemladdru  7618  cauappcvgprlemladdrl  7619  caucvgprlemladdrl  7640  caucvgprprlemaddq  7670  dvdsle  11804